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Mirrors > Home > ILE Home > Th. List > prmuloclemcalc | Unicode version |
Description: Calculations for prmuloc 6664. (Contributed by Jim Kingdon, 9-Dec-2019.) |
Ref | Expression |
---|---|
prmuloclemcalc.ru | |
prmuloclemcalc.udp | |
prmuloclemcalc.axb | |
prmuloclemcalc.pbrx | |
prmuloclemcalc.a | |
prmuloclemcalc.b | |
prmuloclemcalc.d | |
prmuloclemcalc.p | |
prmuloclemcalc.x |
Ref | Expression |
---|---|
prmuloclemcalc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prmuloclemcalc.axb | . . . . . . 7 | |
2 | 1 | oveq2d 5528 | . . . . . 6 |
3 | prmuloclemcalc.ru | . . . . . . . . 9 | |
4 | ltrelnq 6463 | . . . . . . . . . 10 | |
5 | 4 | brel 4392 | . . . . . . . . 9 |
6 | 3, 5 | syl 14 | . . . . . . . 8 |
7 | 6 | simprd 107 | . . . . . . 7 |
8 | prmuloclemcalc.a | . . . . . . 7 | |
9 | prmuloclemcalc.x | . . . . . . 7 | |
10 | distrnqg 6485 | . . . . . . 7 | |
11 | 7, 8, 9, 10 | syl3anc 1135 | . . . . . 6 |
12 | 2, 11 | eqtr3d 2074 | . . . . 5 |
13 | prmuloclemcalc.b | . . . . . . 7 | |
14 | mulcomnqg 6481 | . . . . . . 7 | |
15 | 13, 7, 14 | syl2anc 391 | . . . . . 6 |
16 | prmuloclemcalc.udp | . . . . . . . . . 10 | |
17 | ltmnqi 6501 | . . . . . . . . . 10 | |
18 | 16, 13, 17 | syl2anc 391 | . . . . . . . . 9 |
19 | prmuloclemcalc.d | . . . . . . . . . 10 | |
20 | prmuloclemcalc.p | . . . . . . . . . 10 | |
21 | distrnqg 6485 | . . . . . . . . . 10 | |
22 | 13, 19, 20, 21 | syl3anc 1135 | . . . . . . . . 9 |
23 | 18, 22 | breqtrd 3788 | . . . . . . . 8 |
24 | mulcomnqg 6481 | . . . . . . . . . . 11 | |
25 | 20, 13, 24 | syl2anc 391 | . . . . . . . . . 10 |
26 | prmuloclemcalc.pbrx | . . . . . . . . . 10 | |
27 | 25, 26 | eqbrtrrd 3786 | . . . . . . . . 9 |
28 | mulclnq 6474 | . . . . . . . . . 10 | |
29 | 13, 19, 28 | syl2anc 391 | . . . . . . . . 9 |
30 | ltanqi 6500 | . . . . . . . . 9 | |
31 | 27, 29, 30 | syl2anc 391 | . . . . . . . 8 |
32 | ltsonq 6496 | . . . . . . . . 9 | |
33 | 32, 4 | sotri 4720 | . . . . . . . 8 |
34 | 23, 31, 33 | syl2anc 391 | . . . . . . 7 |
35 | ltmnqi 6501 | . . . . . . . . . 10 | |
36 | 3, 9, 35 | syl2anc 391 | . . . . . . . . 9 |
37 | 6 | simpld 105 | . . . . . . . . . 10 |
38 | mulcomnqg 6481 | . . . . . . . . . 10 | |
39 | 9, 37, 38 | syl2anc 391 | . . . . . . . . 9 |
40 | mulcomnqg 6481 | . . . . . . . . . 10 | |
41 | 9, 7, 40 | syl2anc 391 | . . . . . . . . 9 |
42 | 36, 39, 41 | 3brtr3d 3793 | . . . . . . . 8 |
43 | ltanqi 6500 | . . . . . . . 8 | |
44 | 42, 29, 43 | syl2anc 391 | . . . . . . 7 |
45 | 32, 4 | sotri 4720 | . . . . . . 7 |
46 | 34, 44, 45 | syl2anc 391 | . . . . . 6 |
47 | 15, 46 | eqbrtrrd 3786 | . . . . 5 |
48 | 12, 47 | eqbrtrrd 3786 | . . . 4 |
49 | mulclnq 6474 | . . . . . 6 | |
50 | 7, 8, 49 | syl2anc 391 | . . . . 5 |
51 | mulclnq 6474 | . . . . . 6 | |
52 | 7, 9, 51 | syl2anc 391 | . . . . 5 |
53 | addcomnqg 6479 | . . . . 5 | |
54 | 50, 52, 53 | syl2anc 391 | . . . 4 |
55 | addcomnqg 6479 | . . . . 5 | |
56 | 29, 52, 55 | syl2anc 391 | . . . 4 |
57 | 48, 54, 56 | 3brtr3d 3793 | . . 3 |
58 | ltanqg 6498 | . . . 4 | |
59 | 50, 29, 52, 58 | syl3anc 1135 | . . 3 |
60 | 57, 59 | mpbird 156 | . 2 |
61 | mulcomnqg 6481 | . . 3 | |
62 | 13, 19, 61 | syl2anc 391 | . 2 |
63 | 60, 62 | breqtrd 3788 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wcel 1393 class class class wbr 3764 (class class class)co 5512 cnq 6378 cplq 6380 cmq 6381 cltq 6383 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-ltnqqs 6451 |
This theorem is referenced by: prmuloc 6664 |
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